Robust Hybrid Zero-Order Optimization Algorithms with Acceleration via Averaging in Time

Abstract

We study novel robust zero-order algorithms with acceleration for the solution of real-time optimization problems. In particular, we propose a family of derivative-free dynamics that can be universally modeled as singularly perturbed hybrid dynamical systems with resetting mechanisms. From this family of dynamics, we synthesize four algorithms designed for convex, strongly convex, constrained, and unconstrained zero-order optimization problems. In each case, we establish semi-global practical asymptotic or exponential stability results, and we show how to obtain well-posed discretized algorithms that retain the main properties of the original hybrid dynamics. Given that existing averaging theorems for singularly perturbed hybrid systems are not directly applicable to our setting, we derive a new averaging theorem that relaxes some of the existing assumptions in the literature. This allows us to make a clear link between the KL bounds that characterize the rates of convergence of the hybrid dynamics and their average dynamics. We also show that our results are applicable to non-hybrid zero-order dynamics, thus providing a unifying framework for hybrid and non-hybrid zero-order algorithms based on averaging theory. The superior performance of the hybrid algorithms compared to the traditional schemes that do not incorporate acceleration is illustrated via numerical examples.

Fixed-Time Extremum Seeking

“…2 ) there exists a * > 0 such that for each a ∈ (0, a * ) there exists ε * 1 > 0 such that for each ε 1 ∈ (0, ε * 1 ) the FTGES dynamics ( 9)-( 10) with (u(0), ξ(0), µ(0)) ∈ ({z * } + δB) × ηB × S n generates solutions with unbounded time domain, and each of these solutions satisfies |z(t) − z * | ≤ ν, for all t ≥ T * G . Proof: In order to analyze the FTGES dynamics, we will use averaging tools for dynamical systems that are not necessarily Lipschitz continuous, e.g., [36], [13]. A key feature of these tools is that the KL bound that characterizes the rate of convergence of the "slow state" in a singularly perturbed system is completely characterized by the KL bound of its average system.…”

“…In order to analyze the extremum seeking dynamics considered in this paper, we make use of averaging theorems developed for non-smooth and set-valued systems [36], [37], [13]. This allows us to link the KL bound of the average system with the KL bound that characterizes the properties of the extremum seeking dynamics, thus establishing a semi-global practical fixed-time convergence property.…”

“…However, in contrast to existing smooth extremum seeking controllers, the FTGES dynamics cannot be analyzed using averaging theory and singular perturbation theory for Lipschitz continuous systems. Instead, we will use generalized averaging results developed for set-valued systems and hybrid dynamical systems with possibly non-unique solutions [36], [13], which for ODEs only require continuity of the vector field.…”

“…The main underlaying idea behind extremum seeking control is to induce multiple time scales in the dynamics of the closed-loop system in oder to emulate the behavior of a target nominal optimization algorithm chosen a priori to solve a particular type of optimization problem. Typical target nominal algorithms include gradient descent for convex optimization [8], [9], Newton-like methods [10], Riemannian gradient flows for constrained optimization [11], gradient descent with momentum [12], accelerated gradient descent [13], hybrid gradient descent on manifolds [14], gradient and Newton flows with delays [15], gradient descent with timevarying costs [16], distributed gradient systems for multiagent systems [17], and switched gradient flows modeled as hybrid systems [18]. Other ES approaches based on parameter estimation in adaptive control have also been considered in [19], [20], [21], [22].…”

“…This issue has recently motivated the development of fast ESCs based on Netwon-like flows [10], gradient algorithms with time-invariant momentum [12], discontinuous gradient descent with finite-time stability properties [18,Sec. 6.1], and hybrid gradient flows with momentum resetting [13]. An early use of non-smooth feedback in extremum seeking can be found in control laws (48) and (55) in [28], as well as finitetime (initial condition-dependent) convergence in Figures 2, 4, and 6 of [28].…”